Both bounds have a term, which depends on the energy (first term) and a term which depends on \(g^{-1}h\)  (second term).
Notice, that our bound in eq:symplecticchannelcompare involves \(\Vert g^{-1}h\Vert _\infty \)  and \(\Vert g^{-1}h-1\Vert _2\)  in the second term, whereas the bound from [1]} depends on the square root of these quantities.
In most cases, one is particularly interested in the tightness of the second term. As an example where the second term plays a crucial role, let us consider the study of Trotter errors [2]}.
In this context, \(g=\mathrm {e}^{-t(X+Y)}\)  with \(t\in \mathbb {R}\)  and \(X,Y\in \mathfrak {sp}(2m,\mathbb {R})\)  is some target dynamics, which should be approximated through the Trotter product formula given by \(h=(\mathrm {e}^{-\Omega Xt/L}\mathrm {e}^{-\Omega Yt/L})^L\) .
Here, \(\Vert g^{-1}h-1\Vert _2\)  determines the asymptotic scaling of the Trotter product formula with the number of Trotter steps \(L\) .
It is well-known that the Trotter error asymptotically scales as \(\mathcal {O}(1/L)\)  [2]}, [4]}. This scaling is correctly captured by our bound in eq:symplecticchannelcompare, whereas the bound in [1]} leads to a \(\mathcal {O}(1/\sqrt{L})\)  behaviour.
We compare the two bounds explicitly for an example of such a Trotter problem in fig:symplecticcompare.
<FIGURE>